Two-dimensional bubbles in slow viscous flows

The representation of a biharmonic function in terms of analytic functions is used to transform a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory. The relevant conditions to be satisfied at a free surface, where there is a given surface tension, are derived. A method for dealing with the difficulties of such a free surface is demonstrated by obtaining solutions for a two-dimensional, in viscid bubble in ( a ) a shear flow, and ( b ) a pure straining motion. In both cases the bubble is found to have an elliptical cross-section. The solutions obtained can be shown to be unique only if certain restrictive assumptions are made, and if these are relaxed the same methods may give further solutions. Experiments on three-dimensional inviscid bubbles (Rumscheidt Taylor 1934) demonstrate that angular points appear in the bubble surface, and an analysis is presented to show that such a discontinuity in a two-dimensional free surface is necessarily a genuine cusp and the nature of the flow about such a point is examined.